# ACT Math : How to find the perimeter of a 45/45/90 right isosceles triangle

## Example Questions

### Example Question #1 : How To Find The Perimeter Of A 45/45/90 Right Isosceles Triangle

What is the perimeter of an isosceles right triangle with an hypotenuse of length ?

Explanation:

Your right triangle is a  triangle. It thus looks like this:

Now, you know that you also have a reference triangle for  triangles. This is:

This means that you can set up a ratio to find . It would be:

Your triangle thus could be drawn like this:

Now, notice that you can rationalize the denominator of :

Thus, the perimeter of your figure is:

### Example Question #1 : How To Find The Perimeter Of A 45/45/90 Right Isosceles Triangle

What is the perimeter of an isosceles right triangle with an area of ?

Explanation:

Recall that an isosceles right triangle is also a  triangle. Your reference figure for such a shape is:

or

Now, you know that the area of a triangle is:

For this triangle, though, the base and height are the same. So it is:

Now, we have to be careful, given that our area contains . Let's use , for "side length":

Thus, . Now based on the reference figure above, you can easily see that your triangle is:

### Example Question #201 : Plane Geometry

A tree is  feet tall and is planted in the center of a circular bed with a radius of  feet. If you want to stabalize the tree with ropes going from its midpoint to the border of the bed, how long will each rope measure?

Explanation:

This is a right triangle where the rope is the hypotenuse. One leg is the radius of the circle, 5 feet. The other leg is half of the tree's height, 12 feet. We can now use the Pythagorean Theorem  giving us . If  then .

### Example Question #2 : How To Find The Perimeter Of A 45/45/90 Right Isosceles Triangle

An isosceles right triangle has a hypotenuse of length . What is the perimeter of this triangle, in terms of ?

Explanation:

The ratio of sides to hypotenuse of an isosceles right triangle is always . With this in mind, setting  as our hypotenuse means we must have leg lengths equal to:

Since the perimeter has two of these legs, we just need to multiply this by  and add the result to our hypothesis:

So, our perimeter in terms of  is: